1. Markov ChainsExtra problems

2. Markov Chain Example -- Weather • raining today 40% rain tomorrow • 60% no rain tomorrow • not raining today 20% rain tomorrow • 80% no rain tomorrow Construct a transition graph and a transition matrix.

3. Markov Chain Example – Stock Price • At the end of a given day, the price of a stock is recorded. If the stock has gone up, the probability that it will go up tomorrow is 0.7. If the stock has gone down, the probability that it will go up tomorrow is only 0.5. • Construct a transition graph and a transition matrix.

4. So, X1 = { 3 with probability p 1with probability 1 – p Markov Chain Example -- Gambler’s Ruin At time zero I have X0 = $2, and each day I make a $1 bet. I win with probability p and lose with probability 1– p. I’ll quit if I ever obtain $4 or if I lose all my money. Xt= amount of money I have after the bet on day t. if Xt = 4 then Xt+1 = Xt+2 = • • • = 4 if Xt = 0 then Xt+1 = Xt+2 = • • • = 0. The possible values of Xt is S = { 0, 1, 2, 3, 4 } Construct a transition graph and a transition matrix.

5. Markov Chain Example -- Game of Craps The Game of Craps in Las Vegas plays as follows The player rolls a pair of dice and sums the numbers showing. A total of 7 or 11 on the first rolls wins for the player Where a total of 2, 3, 12 loses Any other number is called the point. The player rolls the dice again. If he/she rolls the point number, she wins If he/she rolls number 7, she loses Any other number requires another roll The game continues until he/she wins or loses. Construct a transition graph and a transition matrix.